Contents
The primary source for these notes is [7] and [4]. However, often we also took inspiration from [5] and [6].
Introduction
The theory of topological vector spaces (TVS), as the name suggests, is a beau- tiful connection between topological and algebraic structures. It has its origin in the need of extending beyond the boundaries of Hilbert and Banach space theory to catch larger classes of spaces and so to better understand their com- mon features eliminating the contest-specific clutter and exploring instead the power of the general structure behind them. The first systematic treatment of these spaces appeared in “Livre V: Espaces vectoriels topologiques (1953)” in the series “´El´ements de math´ematique” by Nicolas Bourbaki. Actually, there was no person called Nicolas Bourbaki but this was just a pseudonym under which a group of mathematicians wrote the above mentioned series of books between 1935 and 1983 with the aim of reformulating the whole mathematics on an extremely formal, rigourous and general basis grounded on set the- ory. The work of the Bourbaki group (officially known as the “Association of collaborators of Nicolas Bourbaki”) greatly influenced the mathematic world and led to the discovery of concepts and terminologies still used today (e.g.
the symbol ;, the notions of injective, surjective, bijective, etc.) The Bour- baki group included several mathematicians connected to the ´Ecole Normale Sup´erieure in Paris such as Henri Cartan, Jean Coulomb, Jean Dieudonn´e, Andr´e Weil, Laurent Schwartz, Jean-Pierre Serre, Alexander Grothendieck.
The latter is surely the name which is most associated to the theory of TVS.
Of course great contributions to this theory were already given before him (e.g. the Banach and Hilbert spaces are examples of TVS), but Alexander Grothendieck was engaged in a completely general approach to the study of these spaces between 1950 and 1955 (see e.g. [1,2]) and collected some among the deepest results on TVS in his Phd thesis [3] written under the supervi- sion of Jean Dieudonn´e and Laurent Schwartz. After his dissertation he said:
“There is nothing more to do, the subject is dead”. Despite this sentence come out of the mouth of a genius, the theory of TVS is far from being dead.
Many aspects are in fact still unknown and the theory lively interacts with several interesting problems which are still currently unsolved!
Chapter 1 Preliminaries
1.1 Topological spaces
1.1.1 The notion of topological space
The topology on a setXis usually defined by specifying its open subsets ofX.
However, in dealing with topological vector spaces, it is often more convenient to define a topology by specifying what the neighbourhoods of each point are.
Definition 1.1.1. Atopology⌧ on a setX is a family of subsets ofX which satisfies the following conditions:
(O1) the empty set ; and the whole X are both in⌧ (O2) ⌧ is closed under finite intersections
(O3) ⌧ is closed under arbitrary unions The pair (X,⌧) is called a topological space.
The setsO 2⌧ are calledopen setsofXand their complementsC =X\O are called closed sets of X. A subset of X may be neither closed nor open, either closed or open, or both. A set that is both closed and open is called a clopen set.
Definition 1.1.2. Let (X,⌧) be a topological space.
• A subfamily B of ⌧ is called a basis if every open set can be written as a union (possibly empty) of sets in B.
• A subfamily X of ⌧ is called a subbasis if the finite intersections of its sets form a basis, i.e. every open set can be written as a union of finite intersections of sets in X.
Therefore, a topology ⌧ on X is completely determined by a basis or a subbasis.
1. Preliminaries
Examples 1.1.3.
a) The family B :={(a, b) : a, b 2Q witha < b} is a basis of the euclidean (or standard) topology on R.
b) The collection S of all semi-infinite intervals of the real line of the forms ( 1, a) and (a,+1), where a2R is not a basis for any topology on R. To show this, suppose it were. Then, for example, ( 1,1) and (0,1) would be in the topology generated by S, being unions of a single basis element, and so their intersection (0,1) would be by the axiom (O2) of topology. But (0,1)clearly cannot be written as a union of elements inS. However, S is a subbasis of the euclidean topology on R.
Proposition 1.1.4. Let X be a set and letB be a collection of subsets of X.
B is a basis for a topology ⌧ on X i↵ the following hold:
1. B covers X, i.e. 8x2X, 9B 2B s.t. x2B.
2. Ifx2B1\B2for someB1, B22B, then9B3 2Bs.t. x2B3✓B1\B2. Proof. (Recap Sheet 1)
Definition 1.1.5. Let (X,⌧) be a topological space and x 2X. A subset U of X is called a neighbourhood of x if it contains an open set containing the point x, i.e. 9O 2⌧ s.t. x2 O ✓U. The family of all neighbourhoods of a pointx2X is denoted byF⌧(x). (In the following, we will omit the subscript whenever there is no ambiguity on the chosen topology.)
In order to define a topology on a set by the family of neighbourhoods of each of its points, it is convenient to introduce the notion of filter. Note that the notion of filter is given on a set which does not need to carry any other structure. Thus this notion is perfectly independent of the topology.
Definition 1.1.6. A filter on a set X is a family F of subsets of X which fulfills the following conditions:
(F1) the empty set ; does not belong to F (F2) F is closed under finite intersections
(F3) any subset of X containing a set in F belongs to F
Definition 1.1.7. A family B of non-empty subsets of a set X is a basis of a filter F on X if
1. B✓F
2. 8A2F,9B2B s.t. B✓A Examples 1.1.8.
a) The family G of all subsets of a set X containing a fixed non-empty sub- set S is a filter and B = {S} is its basis. G is called the principal filter generated by S.
1.1. Topological spaces
b) Given a topological space X and x2X, the family F(x) is a filter.
c) Let S := {xn}n2N be a sequence of points in a set X. Then the family FS := {A ⇢ X : |S\A| < 1} is a filter and it is known as the filter associated to S. For each m 2 N, set Sm := {xn 2 S : n m}. Then B:={Sm :m2N} is a basis for FS.
Proof. (Recap Sheet 1).
Proposition 1.1.9. A family Bof non-empty subsets of a setX is a basis of a filter on X if and only
8 B1, B22B,9B32B s.t. B3 ✓B1\B2. (1.1) Proof.
) Suppose that B is a basis of a filterF on X and let B1, B2 2B. Then, by Definition 1.1.7-1 and (F2), we getB1, B2 2F and so B1\B2 2F. Hence, by Definition 1.1.7-2, there exists B3 2B s.t. B3 ✓B1\B2, i.e. (1.1) holds.
( Suppose that B fulfills (1.1). Then
FB :={A✓X:A◆B for someB 2B} (1.2) is a filter onX (often called thefilter generated by B). In fact, (F1) and (F3) both directly follow from the definition of FB and (F2) holds, because for any A1, A2 2 FB there exist B1, B2 2 B such that B1 ✓ A1 and B2 ✓ A2, and hence (1.1) provides the existence ofB32Bsuch thatB3 ✓B1\B2 ✓A1\A2, which yields A1\A2 2FB. It is totally clear from the definition of FB that Definition1.1.7 is fulfilled and so thatB is basis for the filterFB.
Theorem 1.1.10. Given a topological space X and a point x2X, the filter of neighbourhoods F(x) satisfies the following properties.
(N1) For any A2F(x), x2A.
(N2) For any A2F(x), 9B2F(x): 8y2B,A2F(y).
Viceversa, if for each point x in a setX we are given a filterFx fulfilling the properties (N1) and (N2) then there exists a unique topology ⌧ s.t. for each x2X,Fx is the family of neighbourhoods of x, i.e. Fx⌘F(x),8x2X.
This means that a topology on a set is uniquely determined by the family of neighbourhoods of each of its points.
Proof.
) Let (X,⌧) be a topological space,x2X and F(x) the filter of neighbour- hoods ofx. Then (N1) trivially holds by definition of neighbourhood ofx. To show (N2), let us take A 2 F(x). By the definition of neighbourhood of x,
1. Preliminaries
we know that there exists B 2 ⌧ s.t. x 2 B ✓ A and also that B 2 F(x).
Moreover, since for any y 2 B we have that y 2B ✓A and B 2 ⌧, we can conclude that A2F(y).
( Suppose that for any x in a set X we have a filter Fx fulfilling (N1) and (N2). We aim to show that⌧ := {O ✓X:8 x2O, O 2Fx}1 is the unique topology such that Fx ⌘F⌧(x), 8x2X.
Let us first prove that ⌧ is a topology.
• ; 2 ⌧ by definition of ⌧. Also X 2 ⌧, because for anyx 2X and any A2Fx we clearly have X◆A and so by (F3) X2Fx.
• For anyO1, O2 2⌧, either O1\O2 =; 2⌧ or there exists x2O1\O2. In the latter case, by definition of⌧, we have thatO12FxandO2 2Fx, which imply by (F2) that O1\O2 2Fx and so O1\O22⌧.
• Let U be an arbitrary union of setsUi 2⌧. If U is empty then U 2⌧, otherwise let x2U. Then there exists at least one is.t.x 2Ui and so Ui 2Fx becauseUi 2⌧. ButU ◆Ui, then by (F3) we get that U 2Fx
and so U 2⌧.
It remains to show that ⌧ on X is actually s.t. Fx⌘F⌧(x),8x2X.
• Any U 2 F⌧(x) is a neighbourhood of x and so there exists O 2 ⌧ s.t.
x 2 O ✓ U. Then, by definition of ⌧, we have O 2 Fx and so (F3) implies thatU 2Fx. Hence,F⌧(x)✓Fx.
• Let U 2 Fx and set W := {y 2 U : U 2 Fy} ✓ U. Since x 2 U by (N1), we also have x 2 W. Moreover, if y 2 W then U 2 Fy and so (N2) implies that there existsV 2Fy s.t. 8z2V we haveU 2Fz. This means that z2 W and so V ✓W. Then W 2Fy by (F3). Hence, we have showed that ify 2W thenW 2Fy, i.e. W 2⌧. Summing up, we have just constructed an open set W s.t. x 2W ✓U, i.e. U 2F⌧(x), and so Fx✓F⌧(x).
Note that the non-empty open subsets of any other topology ⌧0 on X such that Fx ⌘F⌧0(x),8x 2X must be identical to the subsetsO of X for which O 2Fx whenever x2O. Hence,⌧0 ⌘⌧.
Remark 1.1.11. The previous proof in particular shows that a subset is open if and only if it is a neighbourhood of each of its points.
Definition 1.1.12. Given a topological space X, a basis B(x) of the filter of neighbourhoods F(x) of x 2 X is called a basis of neighbourhoods of x, i.e.
B(x) is a subset of F(x) s.t. every set in F(x) contains one in B(x). The elements of B(x) are called basic neighbourhoods of x.
1Note that; 2⌧ since a statement that asserts that all members of the empty set have a certain property is always true (vacuous truth).
1.1. Topological spaces
Example 1.1.13. The open sets of a topological space other than the empty set always form a basis of neighbourhoods.
Theorem 1.1.14. Given a topological spaceX and a pointx2X, a basis of open neighbourhoods B(x) satisfies the following properties.
(B1) For any U 2B(x), x2U.
(B2) For any U1, U22B(x), 9U3 2B(x) s.t. U3 ✓U1\U2. (B3) If y2U 2B(x), then 9W 2B(y) s.t. W ✓U.
Viceversa, if for each point x in a setX we are given a collection of subsets Bx fulfilling the properties (B1), (B2) and (B3) then there exists a unique topology ⌧ s.t. for each x 2 X, Bx is a basis of neighbourhoods of x, i.e.
Bx ⌘B(x),8x2X.
Proof. The proof easily follows by using Theorem 1.1.10.
The previous theorem gives a further way of introducing a topology on a set. Indeed, starting from a basis of neighbourhoods of X, we can define a topology on X by setting that a set is open i↵whenever it contains a point it also contains a basic neighbourhood of the point. Thus a topology on a set X is uniquely determined by a basis of neighbourhoods of each of its points.
